Section 3.3: Compound Inequalities

Compound inequalities involve two separate inequalities joined by and or or.

  • AND ( ∩ ): Solution is where both inequalities are true.
  • OR ( ∪ ): Solution is where at least one inequality is true.

Example 1: AND Inequality

Solve \( 1 < x + 2 \le 5 \)

Step 1: Subtract 2 from all parts: \( -1 < x \le 3 \)

Solution: \( -1 < x \le 3 \)

Example 2: OR Inequality

Solve \( x - 3 < -2 \) or \( x + 1 > 4 \)

First inequality: \( x < 1 \)

Second inequality: \( x > 3 \)

Solution: \( x < 1 \) or \( x > 3 \)

Practice Problems

  1. Solve \( 2 \le x + 1 < 6 \)
  2. Solve \( x - 4 > 1 \) or \( x + 2 < 0 \)
  3. Solve \( -3 < 2x - 1 \le 5 \)
  4. Solve \( x + 3 < 2 \) or \( x - 1 \ge 4 \)
  5. Graph \( -2 \le x - 1 < 3 \) on a number line
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